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<div class="fragment"><div class="line"><a name="l00001"></a><span class="lineno">    1</span>&#160;<span class="comment">/* -*- c++ -*- ----------------------------------------------------------</span></div><div class="line"><a name="l00002"></a><span class="lineno">    2</span>&#160;<span class="comment"> *</span></div><div class="line"><a name="l00003"></a><span class="lineno">    3</span>&#160;<span class="comment"> *                    ***       Karamelo       ***</span></div><div class="line"><a name="l00004"></a><span class="lineno">    4</span>&#160;<span class="comment"> *               Parallel Material Point Method Simulator</span></div><div class="line"><a name="l00005"></a><span class="lineno">    5</span>&#160;<span class="comment"> * </span></div><div class="line"><a name="l00006"></a><span class="lineno">    6</span>&#160;<span class="comment"> * Copyright (2019) Alban de Vaucorbeil, alban.devaucorbeil@monash.edu</span></div><div class="line"><a name="l00007"></a><span class="lineno">    7</span>&#160;<span class="comment"> * Materials Science and Engineering, Monash University</span></div><div class="line"><a name="l00008"></a><span class="lineno">    8</span>&#160;<span class="comment"> * Clayton VIC 3800, Australia</span></div><div class="line"><a name="l00009"></a><span class="lineno">    9</span>&#160;<span class="comment"></span></div><div class="line"><a name="l00010"></a><span class="lineno">   10</span>&#160;<span class="comment"> * This software is distributed under the GNU General Public License.</span></div><div class="line"><a name="l00011"></a><span class="lineno">   11</span>&#160;<span class="comment"> *</span></div><div class="line"><a name="l00012"></a><span class="lineno">   12</span>&#160;<span class="comment"> * ----------------------------------------------------------------------- */</span></div><div class="line"><a name="l00013"></a><span class="lineno">   13</span>&#160;</div><div class="line"><a name="l00014"></a><span class="lineno">   14</span>&#160;</div><div class="line"><a name="l00015"></a><span class="lineno">   15</span>&#160;<span class="preprocessor">#ifndef MPM_MATH_H_</span></div><div class="line"><a name="l00016"></a><span class="lineno">   16</span>&#160;<span class="preprocessor">#define MPM_MATH_H_</span></div><div class="line"><a name="l00017"></a><span class="lineno">   17</span>&#160;</div><div class="line"><a name="l00018"></a><span class="lineno">   18</span>&#160;<span class="preprocessor">#include &lt;Eigen/Eigen&gt;</span></div><div class="line"><a name="l00019"></a><span class="lineno">   19</span>&#160;<span class="preprocessor">#include &lt;iostream&gt;</span></div><div class="line"><a name="l00020"></a><span class="lineno">   20</span>&#160;<span class="keyword">using namespace </span><a class="code" href="namespaceEigen.html">Eigen</a>;</div><div class="line"><a name="l00021"></a><span class="lineno">   21</span>&#160;<span class="keyword">using namespace </span><a class="code" href="namespacestd.html">std</a>;</div><div class="line"><a name="l00022"></a><span class="lineno">   22</span>&#160;</div><div class="line"><a name="l00023"></a><span class="lineno">   23</span>&#160;<span class="keyword">namespace </span><a class="code" href="namespaceMPM__Math.html">MPM_Math</a> {</div><div class="line"><a name="l00024"></a><span class="lineno">   24</span>&#160;</div><div class="line"><a name="l00025"></a><span class="lineno">   25</span>&#160;  <span class="comment">/*</span></div><div class="line"><a name="l00026"></a><span class="lineno">   26</span>&#160;<span class="comment"> * deviator of a tensor</span></div><div class="line"><a name="l00027"></a><span class="lineno">   27</span>&#160;<span class="comment"> */</span></div><div class="line"><a name="l00028"></a><span class="lineno">   28</span>&#160;<span class="keyword">static</span> <span class="keyword">inline</span> Matrix3d Deviator(<span class="keyword">const</span> Matrix3d M) {</div><div class="line"><a name="l00029"></a><span class="lineno">   29</span>&#160;        Matrix3d eye;</div><div class="line"><a name="l00030"></a><span class="lineno">   30</span>&#160;        eye.setIdentity();</div><div class="line"><a name="l00031"></a><span class="lineno">   31</span>&#160;        eye *= M.trace() / 3.0;</div><div class="line"><a name="l00032"></a><span class="lineno">   32</span>&#160;        <span class="keywordflow">return</span> M - eye;</div><div class="line"><a name="l00033"></a><span class="lineno">   33</span>&#160;}</div><div class="line"><a name="l00034"></a><span class="lineno">   34</span>&#160;</div><div class="line"><a name="l00035"></a><span class="lineno">   35</span>&#160;<span class="keyword">static</span> <span class="keyword">inline</span> Matrix2d Deviator(<span class="keyword">const</span> Matrix2d M) {</div><div class="line"><a name="l00036"></a><span class="lineno">   36</span>&#160;        Matrix2d eye;</div><div class="line"><a name="l00037"></a><span class="lineno">   37</span>&#160;        eye.setIdentity();</div><div class="line"><a name="l00038"></a><span class="lineno">   38</span>&#160;        eye *= 0.5 * M.trace();</div><div class="line"><a name="l00039"></a><span class="lineno">   39</span>&#160;        <span class="keywordflow">return</span> M - eye;</div><div class="line"><a name="l00040"></a><span class="lineno">   40</span>&#160;}</div><div class="line"><a name="l00041"></a><span class="lineno">   41</span>&#160;<span class="comment">/*</span></div><div class="line"><a name="l00042"></a><span class="lineno">   42</span>&#160;<span class="comment"> * Polar Decomposition M = R * T</span></div><div class="line"><a name="l00043"></a><span class="lineno">   43</span>&#160;<span class="comment"> * where R is a rotation and T a pure translation/stretch matrix.</span></div><div class="line"><a name="l00044"></a><span class="lineno">   44</span>&#160;<span class="comment"> *</span></div><div class="line"><a name="l00045"></a><span class="lineno">   45</span>&#160;<span class="comment"> * The decomposition is achieved using SVD, i.e. M = U S V^T,</span></div><div class="line"><a name="l00046"></a><span class="lineno">   46</span>&#160;<span class="comment"> * where U = R V and S is diagonal.</span></div><div class="line"><a name="l00047"></a><span class="lineno">   47</span>&#160;<span class="comment"> *</span></div><div class="line"><a name="l00048"></a><span class="lineno">   48</span>&#160;<span class="comment"> *</span></div><div class="line"><a name="l00049"></a><span class="lineno">   49</span>&#160;<span class="comment"> * For any physically admissible deformation gradient, the determinant of R must equal +1.</span></div><div class="line"><a name="l00050"></a><span class="lineno">   50</span>&#160;<span class="comment"> * However, scenerios can arise, where the particles interpenetrate and cause inversion, leading to a determinant of R equal to -1.</span></div><div class="line"><a name="l00051"></a><span class="lineno">   51</span>&#160;<span class="comment"> * In this case, the inversion direction is heuristically identified with the eigenvector of the smallest entry of S, which should work for most cases.</span></div><div class="line"><a name="l00052"></a><span class="lineno">   52</span>&#160;<span class="comment"> * The sign of this corresponding eigenvalue is flipped, the original matrix M is recomputed using the flipped S, and the rotation and translation matrices are</span></div><div class="line"><a name="l00053"></a><span class="lineno">   53</span>&#160;<span class="comment"> * obtained again from an SVD. The rotation should proper now, i.e., det(R) = +1.</span></div><div class="line"><a name="l00054"></a><span class="lineno">   54</span>&#160;<span class="comment"> */</span></div><div class="line"><a name="l00055"></a><span class="lineno">   55</span>&#160;</div><div class="line"><a name="l00056"></a><span class="lineno">   56</span>&#160;<span class="keyword">static</span> <span class="keyword">inline</span> <span class="keywordtype">bool</span> PolDec(Matrix3d M, Matrix3d &amp;R, Matrix3d &amp;T, <span class="keywordtype">bool</span> scaleF) {</div><div class="line"><a name="l00057"></a><span class="lineno">   57</span>&#160;</div><div class="line"><a name="l00058"></a><span class="lineno">   58</span>&#160;        JacobiSVD&lt;Matrix3d&gt; svd(M, ComputeFullU | ComputeFullV); <span class="comment">// SVD(A) = U S V*</span></div><div class="line"><a name="l00059"></a><span class="lineno">   59</span>&#160;        Vector3d S_eigenvalues = svd.singularValues();</div><div class="line"><a name="l00060"></a><span class="lineno">   60</span>&#160;        Matrix3d S = svd.singularValues().asDiagonal();</div><div class="line"><a name="l00061"></a><span class="lineno">   61</span>&#160;        Matrix3d U = svd.matrixU();</div><div class="line"><a name="l00062"></a><span class="lineno">   62</span>&#160;        Matrix3d V = svd.matrixV();</div><div class="line"><a name="l00063"></a><span class="lineno">   63</span>&#160;        Matrix3d eye;</div><div class="line"><a name="l00064"></a><span class="lineno">   64</span>&#160;        eye.setIdentity();</div><div class="line"><a name="l00065"></a><span class="lineno">   65</span>&#160;</div><div class="line"><a name="l00066"></a><span class="lineno">   66</span>&#160;        <span class="comment">// now do polar decomposition into M = R * T, where R is rotation</span></div><div class="line"><a name="l00067"></a><span class="lineno">   67</span>&#160;        <span class="comment">// and T is translation matrix</span></div><div class="line"><a name="l00068"></a><span class="lineno">   68</span>&#160;        R = U * V.transpose();</div><div class="line"><a name="l00069"></a><span class="lineno">   69</span>&#160;        T = V * S * V.transpose();</div><div class="line"><a name="l00070"></a><span class="lineno">   70</span>&#160;</div><div class="line"><a name="l00071"></a><span class="lineno">   71</span>&#160;        <span class="keywordflow">if</span> (R.determinant() &lt; 0.0) { <span class="comment">// this is an improper rotation</span></div><div class="line"><a name="l00072"></a><span class="lineno">   72</span>&#160;                <span class="comment">// identify the smallest entry in S and flip its sign</span></div><div class="line"><a name="l00073"></a><span class="lineno">   73</span>&#160;                <span class="keywordtype">int</span> imin;</div><div class="line"><a name="l00074"></a><span class="lineno">   74</span>&#160;                S_eigenvalues.minCoeff(&amp;imin);</div><div class="line"><a name="l00075"></a><span class="lineno">   75</span>&#160;                S(imin, imin) *= -1.0;</div><div class="line"><a name="l00076"></a><span class="lineno">   76</span>&#160;</div><div class="line"><a name="l00077"></a><span class="lineno">   77</span>&#160;                R = M * V * S.inverse() * V.transpose(); <span class="comment">// recompute R using flipped stretch eigenvalues</span></div><div class="line"><a name="l00078"></a><span class="lineno">   78</span>&#160;        }</div><div class="line"><a name="l00079"></a><span class="lineno">   79</span>&#160;</div><div class="line"><a name="l00080"></a><span class="lineno">   80</span>&#160;        <span class="comment">/*</span></div><div class="line"><a name="l00081"></a><span class="lineno">   81</span>&#160;<span class="comment">         * scale S to avoid small principal strains</span></div><div class="line"><a name="l00082"></a><span class="lineno">   82</span>&#160;<span class="comment">         */</span></div><div class="line"><a name="l00083"></a><span class="lineno">   83</span>&#160;</div><div class="line"><a name="l00084"></a><span class="lineno">   84</span>&#160;        <span class="keywordflow">if</span> (scaleF) {</div><div class="line"><a name="l00085"></a><span class="lineno">   85</span>&#160;                <span class="keywordtype">double</span> min = 0.3; <span class="comment">// 0.3^2 = 0.09, should suffice for most problems</span></div><div class="line"><a name="l00086"></a><span class="lineno">   86</span>&#160;                <span class="keywordtype">double</span> max = 2.0;</div><div class="line"><a name="l00087"></a><span class="lineno">   87</span>&#160;                <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; 3; i++) {</div><div class="line"><a name="l00088"></a><span class="lineno">   88</span>&#160;                        <span class="keywordflow">if</span> (S(i, i) &lt; min) {</div><div class="line"><a name="l00089"></a><span class="lineno">   89</span>&#160;                                S(i, i) = min;</div><div class="line"><a name="l00090"></a><span class="lineno">   90</span>&#160;                        } <span class="keywordflow">else</span> <span class="keywordflow">if</span> (S(i, i) &gt; max) {</div><div class="line"><a name="l00091"></a><span class="lineno">   91</span>&#160;                                S(i, i) = max;</div><div class="line"><a name="l00092"></a><span class="lineno">   92</span>&#160;                        }</div><div class="line"><a name="l00093"></a><span class="lineno">   93</span>&#160;                }</div><div class="line"><a name="l00094"></a><span class="lineno">   94</span>&#160;                T = V * S * V.transpose();</div><div class="line"><a name="l00095"></a><span class="lineno">   95</span>&#160;        }</div><div class="line"><a name="l00096"></a><span class="lineno">   96</span>&#160;</div><div class="line"><a name="l00097"></a><span class="lineno">   97</span>&#160;        <span class="keywordflow">if</span> (R.determinant() &gt; 0.0) {</div><div class="line"><a name="l00098"></a><span class="lineno">   98</span>&#160;                <span class="keywordflow">return</span> <span class="keyword">true</span>;</div><div class="line"><a name="l00099"></a><span class="lineno">   99</span>&#160;        } <span class="keywordflow">else</span> {</div><div class="line"><a name="l00100"></a><span class="lineno">  100</span>&#160;                <span class="keywordflow">return</span> <span class="keyword">false</span>;</div><div class="line"><a name="l00101"></a><span class="lineno">  101</span>&#160;        }</div><div class="line"><a name="l00102"></a><span class="lineno">  102</span>&#160;}</div><div class="line"><a name="l00103"></a><span class="lineno">  103</span>&#160;</div><div class="line"><a name="l00104"></a><span class="lineno">  104</span>&#160;<span class="keyword">static</span> <span class="keyword">inline</span> <span class="keywordtype">bool</span> PolDec(Matrix3d M, Matrix3d &amp;R) {</div><div class="line"><a name="l00105"></a><span class="lineno">  105</span>&#160;</div><div class="line"><a name="l00106"></a><span class="lineno">  106</span>&#160;  JacobiSVD&lt;Matrix3d&gt; svd(M, ComputeFullU | ComputeFullV); <span class="comment">// SVD(A) = U S V*</span></div><div class="line"><a name="l00107"></a><span class="lineno">  107</span>&#160;  Vector3d S_eigenvalues = svd.singularValues();</div><div class="line"><a name="l00108"></a><span class="lineno">  108</span>&#160;  Matrix3d S = svd.singularValues().asDiagonal();</div><div class="line"><a name="l00109"></a><span class="lineno">  109</span>&#160;  Matrix3d U = svd.matrixU();</div><div class="line"><a name="l00110"></a><span class="lineno">  110</span>&#160;  Matrix3d V = svd.matrixV();</div><div class="line"><a name="l00111"></a><span class="lineno">  111</span>&#160;  Matrix3d eye;</div><div class="line"><a name="l00112"></a><span class="lineno">  112</span>&#160;  eye.setIdentity();</div><div class="line"><a name="l00113"></a><span class="lineno">  113</span>&#160;</div><div class="line"><a name="l00114"></a><span class="lineno">  114</span>&#160;  <span class="comment">// now do polar decomposition into M = R * T, where R is rotation</span></div><div class="line"><a name="l00115"></a><span class="lineno">  115</span>&#160;  <span class="comment">// and T is translation matrix</span></div><div class="line"><a name="l00116"></a><span class="lineno">  116</span>&#160;  R = U * V.transpose();</div><div class="line"><a name="l00117"></a><span class="lineno">  117</span>&#160;</div><div class="line"><a name="l00118"></a><span class="lineno">  118</span>&#160;  <span class="keywordflow">if</span> (R.determinant() &lt; 0.0) { <span class="comment">// this is an improper rotation</span></div><div class="line"><a name="l00119"></a><span class="lineno">  119</span>&#160;    <span class="comment">// identify the smallest entry in S and flip its sign</span></div><div class="line"><a name="l00120"></a><span class="lineno">  120</span>&#160;    <span class="keywordtype">int</span> imin;</div><div class="line"><a name="l00121"></a><span class="lineno">  121</span>&#160;    S_eigenvalues.minCoeff(&amp;imin);</div><div class="line"><a name="l00122"></a><span class="lineno">  122</span>&#160;    S(imin, imin) *= -1.0;</div><div class="line"><a name="l00123"></a><span class="lineno">  123</span>&#160;</div><div class="line"><a name="l00124"></a><span class="lineno">  124</span>&#160;    R = M * V * S.inverse() *</div><div class="line"><a name="l00125"></a><span class="lineno">  125</span>&#160;        V.transpose(); <span class="comment">// recompute R using flipped stretch eigenvalues</span></div><div class="line"><a name="l00126"></a><span class="lineno">  126</span>&#160;  }</div><div class="line"><a name="l00127"></a><span class="lineno">  127</span>&#160;</div><div class="line"><a name="l00128"></a><span class="lineno">  128</span>&#160;  <span class="keywordflow">if</span> (R.determinant() &gt; 0.0) {</div><div class="line"><a name="l00129"></a><span class="lineno">  129</span>&#160;    <span class="keywordflow">return</span> <span class="keyword">true</span>;</div><div class="line"><a name="l00130"></a><span class="lineno">  130</span>&#160;  } <span class="keywordflow">else</span> {</div><div class="line"><a name="l00131"></a><span class="lineno">  131</span>&#160;    <span class="keywordflow">return</span> <span class="keyword">false</span>;</div><div class="line"><a name="l00132"></a><span class="lineno">  132</span>&#160;  }</div><div class="line"><a name="l00133"></a><span class="lineno">  133</span>&#160;}</div><div class="line"><a name="l00134"></a><span class="lineno">  134</span>&#160;</div><div class="line"><a name="l00135"></a><span class="lineno">  135</span>&#160;<span class="comment">/*</span></div><div class="line"><a name="l00136"></a><span class="lineno">  136</span>&#160;<span class="comment"> * Pseudo-inverse via SVD</span></div><div class="line"><a name="l00137"></a><span class="lineno">  137</span>&#160;<span class="comment"> */</span></div><div class="line"><a name="l00138"></a><span class="lineno">  138</span>&#160;</div><div class="line"><a name="l00139"></a><span class="lineno">  139</span>&#160;<span class="keyword">static</span> <span class="keyword">inline</span> <span class="keywordtype">void</span> pseudo_inverse_SVD(Matrix3d &amp;M) {</div><div class="line"><a name="l00140"></a><span class="lineno">  140</span>&#160;</div><div class="line"><a name="l00141"></a><span class="lineno">  141</span>&#160;        <span class="comment">//JacobiSVD &lt; Matrix3d &gt; svd(M, ComputeFullU | ComputeFullV);</span></div><div class="line"><a name="l00142"></a><span class="lineno">  142</span>&#160;        JacobiSVD&lt;Matrix3d&gt; svd(M, ComputeFullU); <span class="comment">// one Eigevector base is sufficient because matrix is square and symmetric</span></div><div class="line"><a name="l00143"></a><span class="lineno">  143</span>&#160;</div><div class="line"><a name="l00144"></a><span class="lineno">  144</span>&#160;        Vector3d singularValuesInv;</div><div class="line"><a name="l00145"></a><span class="lineno">  145</span>&#160;        Vector3d singularValues = svd.singularValues();</div><div class="line"><a name="l00146"></a><span class="lineno">  146</span>&#160;</div><div class="line"><a name="l00147"></a><span class="lineno">  147</span>&#160;<span class="comment">//cout &lt;&lt; &quot;Here is the matrix V:&quot; &lt;&lt; endl &lt;&lt; V * singularValues.asDiagonal() * U &lt;&lt; endl;</span></div><div class="line"><a name="l00148"></a><span class="lineno">  148</span>&#160;<span class="comment">//cout &lt;&lt; &quot;Its singular values are:&quot; &lt;&lt; endl &lt;&lt; singularValues &lt;&lt; endl;</span></div><div class="line"><a name="l00149"></a><span class="lineno">  149</span>&#160;</div><div class="line"><a name="l00150"></a><span class="lineno">  150</span>&#160;        <span class="keywordtype">double</span> pinvtoler = 1.0e-16; <span class="comment">// 2d machining example goes unstable if this value is increased (1.0e-16).</span></div><div class="line"><a name="l00151"></a><span class="lineno">  151</span>&#160;        <span class="keywordflow">for</span> (<span class="keywordtype">int</span> row = 0; row &lt; 3; row++) {</div><div class="line"><a name="l00152"></a><span class="lineno">  152</span>&#160;                <span class="keywordflow">if</span> (singularValues(row) &gt; pinvtoler) {</div><div class="line"><a name="l00153"></a><span class="lineno">  153</span>&#160;                        singularValuesInv(row) = 1.0 / singularValues(row);</div><div class="line"><a name="l00154"></a><span class="lineno">  154</span>&#160;                } <span class="keywordflow">else</span> {</div><div class="line"><a name="l00155"></a><span class="lineno">  155</span>&#160;                        singularValuesInv(row) = 0.0;</div><div class="line"><a name="l00156"></a><span class="lineno">  156</span>&#160;                }</div><div class="line"><a name="l00157"></a><span class="lineno">  157</span>&#160;        }</div><div class="line"><a name="l00158"></a><span class="lineno">  158</span>&#160;</div><div class="line"><a name="l00159"></a><span class="lineno">  159</span>&#160;        M = svd.matrixU() * singularValuesInv.asDiagonal() * svd.matrixU().transpose();</div><div class="line"><a name="l00160"></a><span class="lineno">  160</span>&#160;</div><div class="line"><a name="l00161"></a><span class="lineno">  161</span>&#160;        <span class="comment">/*</span></div><div class="line"><a name="l00162"></a><span class="lineno">  162</span>&#160;<span class="comment">         * commented version below is for non-symmetric matrices</span></div><div class="line"><a name="l00163"></a><span class="lineno">  163</span>&#160;<span class="comment">         */</span></div><div class="line"><a name="l00164"></a><span class="lineno">  164</span>&#160;</div><div class="line"><a name="l00165"></a><span class="lineno">  165</span>&#160;<span class="comment">//      JacobiSVD &lt; Matrix3d &gt; svd(M, ComputeFullU | ComputeFullV);</span></div><div class="line"><a name="l00166"></a><span class="lineno">  166</span>&#160;<span class="comment">//</span></div><div class="line"><a name="l00167"></a><span class="lineno">  167</span>&#160;<span class="comment">//      Vector3d singularValuesInv;</span></div><div class="line"><a name="l00168"></a><span class="lineno">  168</span>&#160;<span class="comment">//      Vector3d singularValues = svd.singularValues();</span></div><div class="line"><a name="l00169"></a><span class="lineno">  169</span>&#160;<span class="comment">//</span></div><div class="line"><a name="l00170"></a><span class="lineno">  170</span>&#160;<span class="comment">//      //cout &lt;&lt; &quot;Here is the matrix V:&quot; &lt;&lt; endl &lt;&lt; V * singularValues.asDiagonal() * U &lt;&lt; endl;</span></div><div class="line"><a name="l00171"></a><span class="lineno">  171</span>&#160;<span class="comment">//      //cout &lt;&lt; &quot;Its singular values are:&quot; &lt;&lt; endl &lt;&lt; singularValues &lt;&lt; endl;</span></div><div class="line"><a name="l00172"></a><span class="lineno">  172</span>&#160;<span class="comment">//</span></div><div class="line"><a name="l00173"></a><span class="lineno">  173</span>&#160;<span class="comment">//      double pinvtoler = 1.0e-16; // 2d machining example goes unstable if this value is increased (1.0e-16).</span></div><div class="line"><a name="l00174"></a><span class="lineno">  174</span>&#160;<span class="comment">//      for (int row = 0; row &lt; 3; row++) {</span></div><div class="line"><a name="l00175"></a><span class="lineno">  175</span>&#160;<span class="comment">//              if (singularValues(row) &gt; pinvtoler) {</span></div><div class="line"><a name="l00176"></a><span class="lineno">  176</span>&#160;<span class="comment">//                      singularValuesInv(row) = 1.0 / singularValues(row);</span></div><div class="line"><a name="l00177"></a><span class="lineno">  177</span>&#160;<span class="comment">//              } else {</span></div><div class="line"><a name="l00178"></a><span class="lineno">  178</span>&#160;<span class="comment">//                      singularValuesInv(row) = 0.0;</span></div><div class="line"><a name="l00179"></a><span class="lineno">  179</span>&#160;<span class="comment">//              }</span></div><div class="line"><a name="l00180"></a><span class="lineno">  180</span>&#160;<span class="comment">//      }</span></div><div class="line"><a name="l00181"></a><span class="lineno">  181</span>&#160;<span class="comment">//</span></div><div class="line"><a name="l00182"></a><span class="lineno">  182</span>&#160;<span class="comment">//      M = svd.matrixU() * singularValuesInv.asDiagonal() * svd.matrixV().transpose();</span></div><div class="line"><a name="l00183"></a><span class="lineno">  183</span>&#160;}</div><div class="line"><a name="l00184"></a><span class="lineno">  184</span>&#160;</div><div class="line"><a name="l00185"></a><span class="lineno">  185</span>&#160;<span class="comment">/*</span></div><div class="line"><a name="l00186"></a><span class="lineno">  186</span>&#160;<span class="comment"> * specialized SVD for symmetric input matrices.</span></div><div class="line"><a name="l00187"></a><span class="lineno">  187</span>&#160;<span class="comment"> * limits the eigenvalues to the supplied limit value.</span></div><div class="line"><a name="l00188"></a><span class="lineno">  188</span>&#160;<span class="comment"> */</span></div><div class="line"><a name="l00189"></a><span class="lineno">  189</span>&#160;</div><div class="line"><a name="l00190"></a><span class="lineno">  190</span>&#160;<span class="keyword">static</span> <span class="keyword">inline</span> <span class="keywordtype">void</span> pseudo_inverse_SVD_limit_eigenvalue(Matrix3d &amp;M, <span class="keyword">const</span> <span class="keywordtype">double</span> limit) {</div><div class="line"><a name="l00191"></a><span class="lineno">  191</span>&#160;</div><div class="line"><a name="l00192"></a><span class="lineno">  192</span>&#160;        JacobiSVD&lt;Matrix3d&gt; svd(M, ComputeFullU); <span class="comment">// one Eigevector base is sufficient because matrix is square and symmetric</span></div><div class="line"><a name="l00193"></a><span class="lineno">  193</span>&#160;</div><div class="line"><a name="l00194"></a><span class="lineno">  194</span>&#160;        Vector3d singularValuesInv;</div><div class="line"><a name="l00195"></a><span class="lineno">  195</span>&#160;        Vector3d singularValues = svd.singularValues();</div><div class="line"><a name="l00196"></a><span class="lineno">  196</span>&#160;</div><div class="line"><a name="l00197"></a><span class="lineno">  197</span>&#160;        <span class="keywordtype">double</span> pinvtoler = 1.0e-16; <span class="comment">// 2d machining example goes unstable if this value is increased (1.0e-16).</span></div><div class="line"><a name="l00198"></a><span class="lineno">  198</span>&#160;        <span class="keywordflow">for</span> (<span class="keywordtype">int</span> row = 0; row &lt; 3; row++) {</div><div class="line"><a name="l00199"></a><span class="lineno">  199</span>&#160;                <span class="keywordflow">if</span> (singularValues(row) &gt; pinvtoler) {</div><div class="line"><a name="l00200"></a><span class="lineno">  200</span>&#160;                        singularValuesInv(row) = 1.0 / singularValues(row);</div><div class="line"><a name="l00201"></a><span class="lineno">  201</span>&#160;                        <span class="keywordflow">if</span> (singularValuesInv(row) &gt; limit) {</div><div class="line"><a name="l00202"></a><span class="lineno">  202</span>&#160;                                printf(<span class="stringliteral">&quot;eigenvalue is %f\n&quot;</span>, singularValuesInv(row));</div><div class="line"><a name="l00203"></a><span class="lineno">  203</span>&#160;                                singularValuesInv(row) = limit;</div><div class="line"><a name="l00204"></a><span class="lineno">  204</span>&#160;                        }</div><div class="line"><a name="l00205"></a><span class="lineno">  205</span>&#160;                } <span class="keywordflow">else</span> {</div><div class="line"><a name="l00206"></a><span class="lineno">  206</span>&#160;                        singularValuesInv(row) = 0.0;</div><div class="line"><a name="l00207"></a><span class="lineno">  207</span>&#160;                }</div><div class="line"><a name="l00208"></a><span class="lineno">  208</span>&#160;        }</div><div class="line"><a name="l00209"></a><span class="lineno">  209</span>&#160;</div><div class="line"><a name="l00210"></a><span class="lineno">  210</span>&#160;        M = svd.matrixU() * singularValuesInv.asDiagonal() * svd.matrixU().transpose();</div><div class="line"><a name="l00211"></a><span class="lineno">  211</span>&#160;}</div><div class="line"><a name="l00212"></a><span class="lineno">  212</span>&#160;</div><div class="line"><a name="l00213"></a><span class="lineno">  213</span>&#160;<span class="comment">/*</span></div><div class="line"><a name="l00214"></a><span class="lineno">  214</span>&#160;<span class="comment"> * test if two matrices are equal</span></div><div class="line"><a name="l00215"></a><span class="lineno">  215</span>&#160;<span class="comment"> */</span></div><div class="line"><a name="l00216"></a><span class="lineno">  216</span>&#160;<span class="keyword">static</span> <span class="keyword">inline</span> <span class="keywordtype">double</span> TestMatricesEqual(Matrix3d A, Matrix3d B, <span class="keywordtype">double</span> eps) {</div><div class="line"><a name="l00217"></a><span class="lineno">  217</span>&#160;        Matrix3d diff;</div><div class="line"><a name="l00218"></a><span class="lineno">  218</span>&#160;        diff = A - B;</div><div class="line"><a name="l00219"></a><span class="lineno">  219</span>&#160;        <span class="keywordtype">double</span> norm = diff.norm();</div><div class="line"><a name="l00220"></a><span class="lineno">  220</span>&#160;        <span class="keywordflow">if</span> (norm &gt; eps) {</div><div class="line"><a name="l00221"></a><span class="lineno">  221</span>&#160;                printf(<span class="stringliteral">&quot;Matrices A and B are not equal! The L2-norm difference is: %g\n&quot;</span>, norm);</div><div class="line"><a name="l00222"></a><span class="lineno">  222</span>&#160;                cout &lt;&lt; <span class="stringliteral">&quot;Here is matrix A:&quot;</span> &lt;&lt; endl &lt;&lt; A &lt;&lt; endl;</div><div class="line"><a name="l00223"></a><span class="lineno">  223</span>&#160;                cout &lt;&lt; <span class="stringliteral">&quot;Here is matrix B:&quot;</span> &lt;&lt; endl &lt;&lt; B &lt;&lt; endl;</div><div class="line"><a name="l00224"></a><span class="lineno">  224</span>&#160;        }</div><div class="line"><a name="l00225"></a><span class="lineno">  225</span>&#160;        <span class="keywordflow">return</span> norm;</div><div class="line"><a name="l00226"></a><span class="lineno">  226</span>&#160;}</div><div class="line"><a name="l00227"></a><span class="lineno">  227</span>&#160;</div><div class="line"><a name="l00228"></a><span class="lineno">  228</span>&#160;<span class="comment">/* ----------------------------------------------------------------------</span></div><div class="line"><a name="l00229"></a><span class="lineno">  229</span>&#160;<span class="comment"> Limit eigenvalues of a matrix to upper and lower bounds.</span></div><div class="line"><a name="l00230"></a><span class="lineno">  230</span>&#160;<span class="comment"> ------------------------------------------------------------------------- */</span></div><div class="line"><a name="l00231"></a><span class="lineno">  231</span>&#160;</div><div class="line"><a name="l00232"></a><span class="lineno">  232</span>&#160;<span class="keyword">static</span> <span class="keyword">inline</span> Matrix3d LimitEigenvalues(Matrix3d S, <span class="keywordtype">double</span> limitEigenvalue) {</div><div class="line"><a name="l00233"></a><span class="lineno">  233</span>&#160;</div><div class="line"><a name="l00234"></a><span class="lineno">  234</span>&#160;        <span class="comment">/*</span></div><div class="line"><a name="l00235"></a><span class="lineno">  235</span>&#160;<span class="comment">         * compute Eigenvalues of matrix S</span></div><div class="line"><a name="l00236"></a><span class="lineno">  236</span>&#160;<span class="comment">         */</span></div><div class="line"><a name="l00237"></a><span class="lineno">  237</span>&#160;        SelfAdjointEigenSolver&lt;Matrix3d&gt; es;</div><div class="line"><a name="l00238"></a><span class="lineno">  238</span>&#160;        es.compute(S);</div><div class="line"><a name="l00239"></a><span class="lineno">  239</span>&#160;</div><div class="line"><a name="l00240"></a><span class="lineno">  240</span>&#160;        <span class="keywordtype">double</span> max_eigenvalue = es.eigenvalues().maxCoeff();</div><div class="line"><a name="l00241"></a><span class="lineno">  241</span>&#160;        <span class="keywordtype">double</span> min_eigenvalue = es.eigenvalues().minCoeff();</div><div class="line"><a name="l00242"></a><span class="lineno">  242</span>&#160;        <span class="keywordtype">double</span> amax_eigenvalue = fabs(max_eigenvalue);</div><div class="line"><a name="l00243"></a><span class="lineno">  243</span>&#160;        <span class="keywordtype">double</span> amin_eigenvalue = fabs(min_eigenvalue);</div><div class="line"><a name="l00244"></a><span class="lineno">  244</span>&#160;</div><div class="line"><a name="l00245"></a><span class="lineno">  245</span>&#160;        <span class="keywordflow">if</span> ((amax_eigenvalue &gt; limitEigenvalue) || (amin_eigenvalue &gt; limitEigenvalue)) {</div><div class="line"><a name="l00246"></a><span class="lineno">  246</span>&#160;                <span class="keywordflow">if</span> (amax_eigenvalue &gt; amin_eigenvalue) { <span class="comment">// need to scale with max_eigenvalue</span></div><div class="line"><a name="l00247"></a><span class="lineno">  247</span>&#160;                        <span class="keywordtype">double</span> scale = amax_eigenvalue / limitEigenvalue;</div><div class="line"><a name="l00248"></a><span class="lineno">  248</span>&#160;                        Matrix3d V = es.eigenvectors();</div><div class="line"><a name="l00249"></a><span class="lineno">  249</span>&#160;                        Matrix3d S_diag = V.inverse() * S * V; <span class="comment">// diagonalized input matrix</span></div><div class="line"><a name="l00250"></a><span class="lineno">  250</span>&#160;                        S_diag /= scale;</div><div class="line"><a name="l00251"></a><span class="lineno">  251</span>&#160;                        Matrix3d S_scaled = V * S_diag * V.inverse(); <span class="comment">// undiagonalize matrix</span></div><div class="line"><a name="l00252"></a><span class="lineno">  252</span>&#160;                        <span class="keywordflow">return</span> S_scaled;</div><div class="line"><a name="l00253"></a><span class="lineno">  253</span>&#160;                } <span class="keywordflow">else</span> { <span class="comment">// need to scale using min_eigenvalue</span></div><div class="line"><a name="l00254"></a><span class="lineno">  254</span>&#160;                        <span class="keywordtype">double</span> scale = amin_eigenvalue / limitEigenvalue;</div><div class="line"><a name="l00255"></a><span class="lineno">  255</span>&#160;                        Matrix3d V = es.eigenvectors();</div><div class="line"><a name="l00256"></a><span class="lineno">  256</span>&#160;                        Matrix3d S_diag = V.inverse() * S * V; <span class="comment">// diagonalized input matrix</span></div><div class="line"><a name="l00257"></a><span class="lineno">  257</span>&#160;                        S_diag /= scale;</div><div class="line"><a name="l00258"></a><span class="lineno">  258</span>&#160;                        Matrix3d S_scaled = V * S_diag * V.inverse(); <span class="comment">// undiagonalize matrix</span></div><div class="line"><a name="l00259"></a><span class="lineno">  259</span>&#160;                        <span class="keywordflow">return</span> S_scaled;</div><div class="line"><a name="l00260"></a><span class="lineno">  260</span>&#160;                }</div><div class="line"><a name="l00261"></a><span class="lineno">  261</span>&#160;        } <span class="keywordflow">else</span> { <span class="comment">// limiting does not apply</span></div><div class="line"><a name="l00262"></a><span class="lineno">  262</span>&#160;                <span class="keywordflow">return</span> S;</div><div class="line"><a name="l00263"></a><span class="lineno">  263</span>&#160;        }</div><div class="line"><a name="l00264"></a><span class="lineno">  264</span>&#160;}</div><div class="line"><a name="l00265"></a><span class="lineno">  265</span>&#160;</div><div class="line"><a name="l00266"></a><span class="lineno">  266</span>&#160;<span class="keyword">static</span> <span class="keyword">inline</span> <span class="keywordtype">bool</span> LimitMinMaxEigenvalues(Matrix3d &amp;S, <span class="keywordtype">double</span> min, <span class="keywordtype">double</span> max) {</div><div class="line"><a name="l00267"></a><span class="lineno">  267</span>&#160;</div><div class="line"><a name="l00268"></a><span class="lineno">  268</span>&#160;        <span class="comment">/*</span></div><div class="line"><a name="l00269"></a><span class="lineno">  269</span>&#160;<span class="comment">         * compute Eigenvalues of matrix S</span></div><div class="line"><a name="l00270"></a><span class="lineno">  270</span>&#160;<span class="comment">         */</span></div><div class="line"><a name="l00271"></a><span class="lineno">  271</span>&#160;        SelfAdjointEigenSolver&lt;Matrix3d&gt; es;</div><div class="line"><a name="l00272"></a><span class="lineno">  272</span>&#160;        es.compute(S);</div><div class="line"><a name="l00273"></a><span class="lineno">  273</span>&#160;</div><div class="line"><a name="l00274"></a><span class="lineno">  274</span>&#160;        <span class="keywordflow">if</span> ((es.eigenvalues().maxCoeff() &gt; max) || (es.eigenvalues().minCoeff() &lt; min)) {</div><div class="line"><a name="l00275"></a><span class="lineno">  275</span>&#160;                Matrix3d S_diag = es.eigenvalues().asDiagonal();</div><div class="line"><a name="l00276"></a><span class="lineno">  276</span>&#160;                Matrix3d V = es.eigenvectors();</div><div class="line"><a name="l00277"></a><span class="lineno">  277</span>&#160;                <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; 3; i++) {</div><div class="line"><a name="l00278"></a><span class="lineno">  278</span>&#160;                        <span class="keywordflow">if</span> (S_diag(i, i) &lt; min) {</div><div class="line"><a name="l00279"></a><span class="lineno">  279</span>&#160;                                <span class="comment">//printf(&quot;limiting eigenvalue %f --&gt; %f\n&quot;, S_diag(i, i), min);</span></div><div class="line"><a name="l00280"></a><span class="lineno">  280</span>&#160;                                <span class="comment">//printf(&quot;these are the eigenvalues of U: %f %f %f\n&quot;, es.eigenvalues()(0), es.eigenvalues()(1), es.eigenvalues()(2));</span></div><div class="line"><a name="l00281"></a><span class="lineno">  281</span>&#160;                                S_diag(i, i) = min;</div><div class="line"><a name="l00282"></a><span class="lineno">  282</span>&#160;                        } <span class="keywordflow">else</span> <span class="keywordflow">if</span> (S_diag(i, i) &gt; max) {</div><div class="line"><a name="l00283"></a><span class="lineno">  283</span>&#160;                                <span class="comment">//printf(&quot;limiting eigenvalue %f --&gt; %f\n&quot;, S_diag(i, i), max);</span></div><div class="line"><a name="l00284"></a><span class="lineno">  284</span>&#160;                                S_diag(i, i) = max;</div><div class="line"><a name="l00285"></a><span class="lineno">  285</span>&#160;                        }</div><div class="line"><a name="l00286"></a><span class="lineno">  286</span>&#160;                }</div><div class="line"><a name="l00287"></a><span class="lineno">  287</span>&#160;                S = V * S_diag * V.inverse(); <span class="comment">// undiagonalize matrix</span></div><div class="line"><a name="l00288"></a><span class="lineno">  288</span>&#160;                <span class="keywordflow">return</span> <span class="keyword">true</span>;</div><div class="line"><a name="l00289"></a><span class="lineno">  289</span>&#160;        } <span class="keywordflow">else</span> {</div><div class="line"><a name="l00290"></a><span class="lineno">  290</span>&#160;                <span class="keywordflow">return</span> <span class="keyword">false</span>;</div><div class="line"><a name="l00291"></a><span class="lineno">  291</span>&#160;        }</div><div class="line"><a name="l00292"></a><span class="lineno">  292</span>&#160;}</div><div class="line"><a name="l00293"></a><span class="lineno">  293</span>&#160;</div><div class="line"><a name="l00294"></a><span class="lineno">  294</span>&#160;<span class="keyword">static</span> <span class="keyword">inline</span> <span class="keywordtype">void</span> reconstruct_rank_deficient_shape_matrix(Matrix3d &amp;K) {</div><div class="line"><a name="l00295"></a><span class="lineno">  295</span>&#160;</div><div class="line"><a name="l00296"></a><span class="lineno">  296</span>&#160;        JacobiSVD&lt;Matrix3d&gt; svd(K, ComputeFullU | ComputeFullV);</div><div class="line"><a name="l00297"></a><span class="lineno">  297</span>&#160;        Vector3d singularValues = svd.singularValues();</div><div class="line"><a name="l00298"></a><span class="lineno">  298</span>&#160;</div><div class="line"><a name="l00299"></a><span class="lineno">  299</span>&#160;        <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; 3; i++) {</div><div class="line"><a name="l00300"></a><span class="lineno">  300</span>&#160;                <span class="keywordflow">if</span> (singularValues(i) &lt; 1.0e-8) {</div><div class="line"><a name="l00301"></a><span class="lineno">  301</span>&#160;                        singularValues(i) = 1.0;</div><div class="line"><a name="l00302"></a><span class="lineno">  302</span>&#160;                }</div><div class="line"><a name="l00303"></a><span class="lineno">  303</span>&#160;        }</div><div class="line"><a name="l00304"></a><span class="lineno">  304</span>&#160;</div><div class="line"><a name="l00305"></a><span class="lineno">  305</span>&#160;<span class="comment">//              int imin;</span></div><div class="line"><a name="l00306"></a><span class="lineno">  306</span>&#160;<span class="comment">//              double minev = singularValues.minCoeff(&amp;imin);</span></div><div class="line"><a name="l00307"></a><span class="lineno">  307</span>&#160;<span class="comment">//</span></div><div class="line"><a name="l00308"></a><span class="lineno">  308</span>&#160;<span class="comment">//              printf(&quot;min eigenvalue=%f has index %d\n&quot;, minev, imin);</span></div><div class="line"><a name="l00309"></a><span class="lineno">  309</span>&#160;<span class="comment">//              Vector3d singularVec = U.col(0).cross(U.col(1));</span></div><div class="line"><a name="l00310"></a><span class="lineno">  310</span>&#160;<span class="comment">//              cout &lt;&lt; &quot;the eigenvalues are &quot; &lt;&lt; endl &lt;&lt; singularValues &lt;&lt; endl;</span></div><div class="line"><a name="l00311"></a><span class="lineno">  311</span>&#160;<span class="comment">//              cout &lt;&lt; &quot;the singular vector is &quot; &lt;&lt; endl &lt;&lt; singularVec &lt;&lt; endl;</span></div><div class="line"><a name="l00312"></a><span class="lineno">  312</span>&#160;<span class="comment">//</span></div><div class="line"><a name="l00313"></a><span class="lineno">  313</span>&#160;<span class="comment">//              // reconstruct original K</span></div><div class="line"><a name="l00314"></a><span class="lineno">  314</span>&#160;<span class="comment">//</span></div><div class="line"><a name="l00315"></a><span class="lineno">  315</span>&#160;<span class="comment">//              singularValues(2) = 1.0;</span></div><div class="line"><a name="l00316"></a><span class="lineno">  316</span>&#160;</div><div class="line"><a name="l00317"></a><span class="lineno">  317</span>&#160;        K = svd.matrixU() * singularValues.asDiagonal() * svd.matrixV().transpose();</div><div class="line"><a name="l00318"></a><span class="lineno">  318</span>&#160;        <span class="comment">//cout &lt;&lt; &quot;the reconstructed K is &quot; &lt;&lt; endl &lt;&lt; K &lt;&lt; endl;</span></div><div class="line"><a name="l00319"></a><span class="lineno">  319</span>&#160;        <span class="comment">//exit(1);</span></div><div class="line"><a name="l00320"></a><span class="lineno">  320</span>&#160;}</div><div class="line"><a name="l00321"></a><span class="lineno">  321</span>&#160;</div><div class="line"><a name="l00322"></a><span class="lineno">  322</span>&#160;<span class="comment">/* ----------------------------------------------------------------------</span></div><div class="line"><a name="l00323"></a><span class="lineno">  323</span>&#160;<span class="comment"> helper functions for crack_exclude</span></div><div class="line"><a name="l00324"></a><span class="lineno">  324</span>&#160;<span class="comment"> ------------------------------------------------------------------------- */</span></div><div class="line"><a name="l00325"></a><span class="lineno">  325</span>&#160;<span class="keyword">static</span> <span class="keyword">inline</span> <span class="keywordtype">bool</span> IsOnSegment(<span class="keywordtype">double</span> xi, <span class="keywordtype">double</span> yi, <span class="keywordtype">double</span> xj, <span class="keywordtype">double</span> yj, <span class="keywordtype">double</span> xk, <span class="keywordtype">double</span> yk) {</div><div class="line"><a name="l00326"></a><span class="lineno">  326</span>&#160;        <span class="keywordflow">return</span> (xi &lt;= xk || xj &lt;= xk) &amp;&amp; (xk &lt;= xi || xk &lt;= xj) &amp;&amp; (yi &lt;= yk || yj &lt;= yk) &amp;&amp; (yk &lt;= yi || yk &lt;= yj);</div><div class="line"><a name="l00327"></a><span class="lineno">  327</span>&#160;}</div><div class="line"><a name="l00328"></a><span class="lineno">  328</span>&#160;</div><div class="line"><a name="l00329"></a><span class="lineno">  329</span>&#160;<span class="keyword">static</span> <span class="keyword">inline</span> <span class="keywordtype">char</span> ComputeDirection(<span class="keywordtype">double</span> xi, <span class="keywordtype">double</span> yi, <span class="keywordtype">double</span> xj, <span class="keywordtype">double</span> yj, <span class="keywordtype">double</span> xk, <span class="keywordtype">double</span> yk) {</div><div class="line"><a name="l00330"></a><span class="lineno">  330</span>&#160;        <span class="keywordtype">double</span> a = (xk - xi) * (yj - yi);</div><div class="line"><a name="l00331"></a><span class="lineno">  331</span>&#160;        <span class="keywordtype">double</span> b = (xj - xi) * (yk - yi);</div><div class="line"><a name="l00332"></a><span class="lineno">  332</span>&#160;        <span class="keywordflow">return</span> a &lt; b ? -1.0 : a &gt; b ? 1.0 : 0;</div><div class="line"><a name="l00333"></a><span class="lineno">  333</span>&#160;}</div><div class="line"><a name="l00334"></a><span class="lineno">  334</span>&#160;</div><div class="line"><a name="l00336"></a><span class="lineno">  336</span>&#160;<span class="keyword">static</span> <span class="keyword">inline</span> <span class="keywordtype">bool</span> DoLineSegmentsIntersect(<span class="keywordtype">double</span> x1, <span class="keywordtype">double</span> y1, <span class="keywordtype">double</span> x2, <span class="keywordtype">double</span> y2, <span class="keywordtype">double</span> x3, <span class="keywordtype">double</span> y3, <span class="keywordtype">double</span> x4, <span class="keywordtype">double</span> y4) {</div><div class="line"><a name="l00337"></a><span class="lineno">  337</span>&#160;        <span class="keywordtype">char</span> d1 = ComputeDirection(x3, y3, x4, y4, x1, y1);</div><div class="line"><a name="l00338"></a><span class="lineno">  338</span>&#160;        <span class="keywordtype">char</span> d2 = ComputeDirection(x3, y3, x4, y4, x2, y2);</div><div class="line"><a name="l00339"></a><span class="lineno">  339</span>&#160;        <span class="keywordtype">char</span> d3 = ComputeDirection(x1, y1, x2, y2, x3, y3);</div><div class="line"><a name="l00340"></a><span class="lineno">  340</span>&#160;        <span class="keywordtype">char</span> d4 = ComputeDirection(x1, y1, x2, y2, x4, y4);</div><div class="line"><a name="l00341"></a><span class="lineno">  341</span>&#160;        <span class="keywordflow">return</span> (((d1 &gt; 0 &amp;&amp; d2 &lt; 0) || (d1 &lt; 0 &amp;&amp; d2 &gt; 0)) &amp;&amp; ((d3 &gt; 0 &amp;&amp; d4 &lt; 0) || (d3 &lt; 0 &amp;&amp; d4 &gt; 0)))</div><div class="line"><a name="l00342"></a><span class="lineno">  342</span>&#160;                        || (d1 == 0 &amp;&amp; IsOnSegment(x3, y3, x4, y4, x1, y1)) || (d2 == 0 &amp;&amp; IsOnSegment(x3, y3, x4, y4, x2, y2))</div><div class="line"><a name="l00343"></a><span class="lineno">  343</span>&#160;                        || (d3 == 0 &amp;&amp; IsOnSegment(x1, y1, x2, y2, x3, y3)) || (d4 == 0 &amp;&amp; IsOnSegment(x1, y1, x2, y2, x4, y4));</div><div class="line"><a name="l00344"></a><span class="lineno">  344</span>&#160;}</div><div class="line"><a name="l00345"></a><span class="lineno">  345</span>&#160;</div><div class="line"><a name="l00346"></a><span class="lineno">  346</span>&#160;<span class="comment">/*</span></div><div class="line"><a name="l00347"></a><span class="lineno">  347</span>&#160;<span class="comment"> * Pseudo-inverse via SVD for a 4d matrix</span></div><div class="line"><a name="l00348"></a><span class="lineno">  348</span>&#160;<span class="comment"> */</span></div><div class="line"><a name="l00349"></a><span class="lineno">  349</span>&#160;</div><div class="line"><a name="l00350"></a><span class="lineno">  350</span>&#160;<span class="keyword">static</span> <span class="keyword">inline</span> <span class="keywordtype">void</span> pinv4(Matrix4d &amp;M) {</div><div class="line"><a name="l00351"></a><span class="lineno">  351</span>&#160;</div><div class="line"><a name="l00352"></a><span class="lineno">  352</span>&#160;        JacobiSVD &lt; Matrix4d &gt; svd(M, ComputeFullU | ComputeFullV);</div><div class="line"><a name="l00353"></a><span class="lineno">  353</span>&#160;        <span class="comment">//JacobiSVD&lt;Matrix4d&gt; svd(M, ComputeFullU); // one Eigevector base is sufficient because matrix is square and symmetric</span></div><div class="line"><a name="l00354"></a><span class="lineno">  354</span>&#160;</div><div class="line"><a name="l00355"></a><span class="lineno">  355</span>&#160;        Vector4d singularValuesInv;</div><div class="line"><a name="l00356"></a><span class="lineno">  356</span>&#160;        Vector4d singularValues = svd.singularValues();</div><div class="line"><a name="l00357"></a><span class="lineno">  357</span>&#160;</div><div class="line"><a name="l00358"></a><span class="lineno">  358</span>&#160;<span class="comment">//cout &lt;&lt; &quot;Here is the matrix V:&quot; &lt;&lt; endl &lt;&lt; svd.MatrixV * singularValues.asDiagonal() * U &lt;&lt; endl;</span></div><div class="line"><a name="l00359"></a><span class="lineno">  359</span>&#160;cout &lt;&lt; <span class="stringliteral">&quot;Its singular values are:&quot;</span> &lt;&lt; endl &lt;&lt; singularValues &lt;&lt; endl;</div><div class="line"><a name="l00360"></a><span class="lineno">  360</span>&#160;</div><div class="line"><a name="l00361"></a><span class="lineno">  361</span>&#160;        <span class="keywordtype">double</span> pinvtoler = 1.0e-3; <span class="comment">// 2d machining example goes unstable if this value is increased (1.0e-16).</span></div><div class="line"><a name="l00362"></a><span class="lineno">  362</span>&#160;        <span class="keywordflow">for</span> (<span class="keywordtype">int</span> row = 0; row &lt; 4; row++) {</div><div class="line"><a name="l00363"></a><span class="lineno">  363</span>&#160;                <span class="keywordflow">if</span> (singularValues(row) &gt; pinvtoler) {</div><div class="line"><a name="l00364"></a><span class="lineno">  364</span>&#160;                        singularValuesInv(row) = 1.0 / singularValues(row);</div><div class="line"><a name="l00365"></a><span class="lineno">  365</span>&#160;                } <span class="keywordflow">else</span> {</div><div class="line"><a name="l00366"></a><span class="lineno">  366</span>&#160;                        singularValuesInv(row) = 0.0;</div><div class="line"><a name="l00367"></a><span class="lineno">  367</span>&#160;                }</div><div class="line"><a name="l00368"></a><span class="lineno">  368</span>&#160;        }</div><div class="line"><a name="l00369"></a><span class="lineno">  369</span>&#160;</div><div class="line"><a name="l00370"></a><span class="lineno">  370</span>&#160;        M = svd.matrixV() * singularValuesInv.asDiagonal() * svd.matrixU().transpose();</div><div class="line"><a name="l00371"></a><span class="lineno">  371</span>&#160;}</div><div class="line"><a name="l00372"></a><span class="lineno">  372</span>&#160;</div><div class="line"><a name="l00373"></a><span class="lineno">  373</span>&#160;<span class="comment">/* circle_circle_intersection() *</span></div><div class="line"><a name="l00374"></a><span class="lineno">  374</span>&#160;<span class="comment"> * Determine the points where 2 circles in a common plane intersect.</span></div><div class="line"><a name="l00375"></a><span class="lineno">  375</span>&#160;<span class="comment"> *</span></div><div class="line"><a name="l00376"></a><span class="lineno">  376</span>&#160;<span class="comment"> * int circle_circle_intersection(</span></div><div class="line"><a name="l00377"></a><span class="lineno">  377</span>&#160;<span class="comment"> *                                // center and radius of 1st circle</span></div><div class="line"><a name="l00378"></a><span class="lineno">  378</span>&#160;<span class="comment"> *                                double x0, double y0, double r0,</span></div><div class="line"><a name="l00379"></a><span class="lineno">  379</span>&#160;<span class="comment"> *                                // center and radius of 2nd circle</span></div><div class="line"><a name="l00380"></a><span class="lineno">  380</span>&#160;<span class="comment"> *                                double x1, double y1, double r1,</span></div><div class="line"><a name="l00381"></a><span class="lineno">  381</span>&#160;<span class="comment"> *                                // 1st intersection point</span></div><div class="line"><a name="l00382"></a><span class="lineno">  382</span>&#160;<span class="comment"> *                                double *xi, double *yi,</span></div><div class="line"><a name="l00383"></a><span class="lineno">  383</span>&#160;<span class="comment"> *                                // 2nd intersection point</span></div><div class="line"><a name="l00384"></a><span class="lineno">  384</span>&#160;<span class="comment"> *                                double *xi_prime, double *yi_prime)</span></div><div class="line"><a name="l00385"></a><span class="lineno">  385</span>&#160;<span class="comment"> *</span></div><div class="line"><a name="l00386"></a><span class="lineno">  386</span>&#160;<span class="comment"> * This is a public domain work. 3/26/2005 Tim Voght</span></div><div class="line"><a name="l00387"></a><span class="lineno">  387</span>&#160;<span class="comment"> *</span></div><div class="line"><a name="l00388"></a><span class="lineno">  388</span>&#160;<span class="comment"> */</span></div><div class="line"><a name="l00389"></a><span class="lineno">  389</span>&#160;<span class="comment">//#include &lt;stdio.h&gt;</span></div><div class="line"><a name="l00390"></a><span class="lineno">  390</span>&#160;<span class="comment">//#include &lt;math.h&gt;</span></div><div class="line"><a name="l00391"></a><span class="lineno">  391</span>&#160;</div><div class="line"><a name="l00392"></a><span class="lineno">  392</span>&#160;<span class="keyword">static</span> <span class="keywordtype">int</span> circle_circle_intersection(<span class="keywordtype">double</span> x0, <span class="keywordtype">double</span> y0, <span class="keywordtype">double</span> r0,</div><div class="line"><a name="l00393"></a><span class="lineno">  393</span>&#160;                               <span class="keywordtype">double</span> x1, <span class="keywordtype">double</span> y1, <span class="keywordtype">double</span> r1,</div><div class="line"><a name="l00394"></a><span class="lineno">  394</span>&#160;                               <span class="keywordtype">double</span> &amp;xi, <span class="keywordtype">double</span> &amp;yi,</div><div class="line"><a name="l00395"></a><span class="lineno">  395</span>&#160;                               <span class="keywordtype">double</span> &amp;xi_prime, <span class="keywordtype">double</span> &amp;yi_prime)</div><div class="line"><a name="l00396"></a><span class="lineno">  396</span>&#160;{</div><div class="line"><a name="l00397"></a><span class="lineno">  397</span>&#160;  <span class="keywordtype">double</span> a, dx, dy, d, h, rx, ry;</div><div class="line"><a name="l00398"></a><span class="lineno">  398</span>&#160;  <span class="keywordtype">double</span> x2, y2;</div><div class="line"><a name="l00399"></a><span class="lineno">  399</span>&#160;</div><div class="line"><a name="l00400"></a><span class="lineno">  400</span>&#160;  <span class="comment">/* dx and dy are the vertical and horizontal distances between</span></div><div class="line"><a name="l00401"></a><span class="lineno">  401</span>&#160;<span class="comment">   * the circle centers.</span></div><div class="line"><a name="l00402"></a><span class="lineno">  402</span>&#160;<span class="comment">   */</span></div><div class="line"><a name="l00403"></a><span class="lineno">  403</span>&#160;  dx = x1 - x0;</div><div class="line"><a name="l00404"></a><span class="lineno">  404</span>&#160;  dy = y1 - y0;</div><div class="line"><a name="l00405"></a><span class="lineno">  405</span>&#160;</div><div class="line"><a name="l00406"></a><span class="lineno">  406</span>&#160;  <span class="comment">/* Determine the straight-line distance between the centers. */</span></div><div class="line"><a name="l00407"></a><span class="lineno">  407</span>&#160;  <span class="comment">//d = sqrt((dy*dy) + (dx*dx));</span></div><div class="line"><a name="l00408"></a><span class="lineno">  408</span>&#160;  d = hypot(dx,dy); <span class="comment">// Suggested by Keith Briggs</span></div><div class="line"><a name="l00409"></a><span class="lineno">  409</span>&#160;</div><div class="line"><a name="l00410"></a><span class="lineno">  410</span>&#160;  <span class="comment">/* Check for solvability. */</span></div><div class="line"><a name="l00411"></a><span class="lineno">  411</span>&#160;  <span class="keywordflow">if</span> (d &gt; (r0 + r1))</div><div class="line"><a name="l00412"></a><span class="lineno">  412</span>&#160;  {</div><div class="line"><a name="l00413"></a><span class="lineno">  413</span>&#160;    <span class="comment">/* no solution. circles do not intersect. */</span></div><div class="line"><a name="l00414"></a><span class="lineno">  414</span>&#160;    <span class="keywordflow">return</span> 0;</div><div class="line"><a name="l00415"></a><span class="lineno">  415</span>&#160;  }</div><div class="line"><a name="l00416"></a><span class="lineno">  416</span>&#160;  <span class="keywordflow">if</span> (d &lt; fabs(r0 - r1))</div><div class="line"><a name="l00417"></a><span class="lineno">  417</span>&#160;  {</div><div class="line"><a name="l00418"></a><span class="lineno">  418</span>&#160;    <span class="comment">/* no solution. one circle is contained in the other */</span></div><div class="line"><a name="l00419"></a><span class="lineno">  419</span>&#160;    <span class="keywordflow">return</span> 0;</div><div class="line"><a name="l00420"></a><span class="lineno">  420</span>&#160;  }</div><div class="line"><a name="l00421"></a><span class="lineno">  421</span>&#160;</div><div class="line"><a name="l00422"></a><span class="lineno">  422</span>&#160;  <span class="comment">/* &#39;point 2&#39; is the point where the line through the circle</span></div><div class="line"><a name="l00423"></a><span class="lineno">  423</span>&#160;<span class="comment">   * intersection points crosses the line between the circle</span></div><div class="line"><a name="l00424"></a><span class="lineno">  424</span>&#160;<span class="comment">   * centers.</span></div><div class="line"><a name="l00425"></a><span class="lineno">  425</span>&#160;<span class="comment">   */</span></div><div class="line"><a name="l00426"></a><span class="lineno">  426</span>&#160;</div><div class="line"><a name="l00427"></a><span class="lineno">  427</span>&#160;  <span class="comment">/* Determine the distance from point 0 to point 2. */</span></div><div class="line"><a name="l00428"></a><span class="lineno">  428</span>&#160;  a = ((r0*r0) - (r1*r1) + (d*d)) / (2.0 * d) ;</div><div class="line"><a name="l00429"></a><span class="lineno">  429</span>&#160;</div><div class="line"><a name="l00430"></a><span class="lineno">  430</span>&#160;  <span class="comment">/* Determine the coordinates of point 2. */</span></div><div class="line"><a name="l00431"></a><span class="lineno">  431</span>&#160;  x2 = x0 + (dx * a/d);</div><div class="line"><a name="l00432"></a><span class="lineno">  432</span>&#160;  y2 = y0 + (dy * a/d);</div><div class="line"><a name="l00433"></a><span class="lineno">  433</span>&#160;</div><div class="line"><a name="l00434"></a><span class="lineno">  434</span>&#160;  <span class="comment">/* Determine the distance from point 2 to either of the</span></div><div class="line"><a name="l00435"></a><span class="lineno">  435</span>&#160;<span class="comment">   * intersection points.</span></div><div class="line"><a name="l00436"></a><span class="lineno">  436</span>&#160;<span class="comment">   */</span></div><div class="line"><a name="l00437"></a><span class="lineno">  437</span>&#160;  h = sqrt((r0*r0) - (a*a));</div><div class="line"><a name="l00438"></a><span class="lineno">  438</span>&#160;</div><div class="line"><a name="l00439"></a><span class="lineno">  439</span>&#160;  <span class="comment">/* Now determine the offsets of the intersection points from</span></div><div class="line"><a name="l00440"></a><span class="lineno">  440</span>&#160;<span class="comment">   * point 2.</span></div><div class="line"><a name="l00441"></a><span class="lineno">  441</span>&#160;<span class="comment">   */</span></div><div class="line"><a name="l00442"></a><span class="lineno">  442</span>&#160;  rx = -dy * (h/d);</div><div class="line"><a name="l00443"></a><span class="lineno">  443</span>&#160;  ry = dx * (h/d);</div><div class="line"><a name="l00444"></a><span class="lineno">  444</span>&#160;</div><div class="line"><a name="l00445"></a><span class="lineno">  445</span>&#160;  <span class="comment">/* Determine the absolute intersection points. */</span></div><div class="line"><a name="l00446"></a><span class="lineno">  446</span>&#160;  xi = x2 + rx;</div><div class="line"><a name="l00447"></a><span class="lineno">  447</span>&#160;  xi_prime = x2 - rx;</div><div class="line"><a name="l00448"></a><span class="lineno">  448</span>&#160;  yi = y2 + ry;</div><div class="line"><a name="l00449"></a><span class="lineno">  449</span>&#160;  yi_prime = y2 - ry;</div><div class="line"><a name="l00450"></a><span class="lineno">  450</span>&#160;</div><div class="line"><a name="l00451"></a><span class="lineno">  451</span>&#160;  <span class="keywordflow">return</span> 1;</div><div class="line"><a name="l00452"></a><span class="lineno">  452</span>&#160;}</div><div class="line"><a name="l00453"></a><span class="lineno">  453</span>&#160;</div><div class="line"><a name="l00454"></a><span class="lineno">  454</span>&#160;<span class="keyword">static</span> <span class="keyword">inline</span> <span class="keywordtype">double</span> clamp(<span class="keywordtype">double</span> x, <span class="keywordtype">double</span> lowerlimit, <span class="keywordtype">double</span> upperlimit)</div><div class="line"><a name="l00455"></a><span class="lineno">  455</span>&#160;{</div><div class="line"><a name="l00456"></a><span class="lineno">  456</span>&#160;    <span class="keywordflow">if</span> (x &lt; lowerlimit) x = lowerlimit;</div><div class="line"><a name="l00457"></a><span class="lineno">  457</span>&#160;    <span class="keywordflow">if</span> (x &gt; upperlimit) x = upperlimit;</div><div class="line"><a name="l00458"></a><span class="lineno">  458</span>&#160;    <span class="keywordflow">return</span> x;</div><div class="line"><a name="l00459"></a><span class="lineno">  459</span>&#160;}</div><div class="line"><a name="l00460"></a><span class="lineno">  460</span>&#160;</div><div class="line"><a name="l00461"></a><span class="lineno">  461</span>&#160;<span class="keyword">static</span> <span class="keyword">inline</span> <span class="keywordtype">double</span> smoothstep(<span class="keywordtype">double</span> edge0, <span class="keywordtype">double</span> edge1, <span class="keywordtype">double</span> x)</div><div class="line"><a name="l00462"></a><span class="lineno">  462</span>&#160;{</div><div class="line"><a name="l00463"></a><span class="lineno">  463</span>&#160;    <span class="comment">// Scale, bias and saturate x to 0..1 range</span></div><div class="line"><a name="l00464"></a><span class="lineno">  464</span>&#160;    x = clamp((x - edge0)/(edge1 - edge0), 0.0, 1.0);</div><div class="line"><a name="l00465"></a><span class="lineno">  465</span>&#160;    <span class="comment">// Evaluate polynomial</span></div><div class="line"><a name="l00466"></a><span class="lineno">  466</span>&#160;    <span class="keywordflow">return</span> x*x*(3 - 2*x);</div><div class="line"><a name="l00467"></a><span class="lineno">  467</span>&#160;}</div><div class="line"><a name="l00468"></a><span class="lineno">  468</span>&#160;</div><div class="line"><a name="l00469"></a><span class="lineno">  469</span>&#160;}</div><div class="line"><a name="l00470"></a><span class="lineno">  470</span>&#160;</div><div class="line"><a name="l00471"></a><span class="lineno">  471</span>&#160;<span class="preprocessor">#endif </span><span class="comment">/* SMD_MATH_H_ */</span><span class="preprocessor"></span></div><div class="ttc" id="namespaceEigen_html"><div class="ttname"><a href="namespaceEigen.html">Eigen</a></div></div>
<div class="ttc" id="namespacestd_html"><div class="ttname"><a href="namespacestd.html">std</a></div></div>
<div class="ttc" id="namespaceMPM__Math_html"><div class="ttname"><a href="namespaceMPM__Math.html">MPM_Math</a></div><div class="ttdef"><b>Definition:</b> mpm_math.h:23</div></div>
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